Question

If the function f is continuous at x = 2 and x = 4 then find the values of a and b.

Where f(x) = x² + ax + b, x < 2  

= 3x + 2, 2 ≤ x ≤ 4 

=  2ax + 5b, 4 < x 

Answer 1

Given f is continuous at x = 2

∴ `lim_(x->2^-) "f(x)" = lim_(x->2^+) "f(x)" = "f"(2)`

Consider

`lim_(x->2^-) "f(x)" = lim_(x->2^+) "f(x)"`  

`=> lim_(x->2) ["x"^2 + "ax" +"b"] = lim_(x->2) ["3x + 2"]`  

⇒ (2)² + a(2) + b= 3(2) + 2 

⇒  4 + 2a + b = 6 + 2 

⇒ 2a + b = 4 ....... (i) 

Also f is continuous at x= 4 

`therefore lim_(x->4^-) "f(x)" = lim_(x->4^+) "f(x)"`

`=> lim_(x->4) ("3x + 2") = lim_(x->4^+) ["2ax + 5b"]`

⇒ 3(4) + 2 = 2a(4) +5b 

⇒ 12 + 2 = 8a + 5b 

⇒ 8a + 5b = 14       .......(ii)

Solving (i) and (ii)

Multiplying equation (i) by 4 · 

8a+ 4b = 16

8a + 5b = 14 

subtracting - b = 2 

∴ b = -2

putting b = -2 in equation (i) 

2a - 2 = 4

⇒  2a =4 + 2 

∴ a = 3